Homogenized Spectral Problems for Exactly Solvable Operators: Asymptotics of Polynomial Eigenfunctions
Julius Borcea, Rikard Bøgvad, Boris Zalmanovich Shapiro · Publications of the Research Institute for Mathematical Sciences · 2009
Consider a homogenized spectral pencil of exactly solvable linear differential operators T_λ = \sum^k_{i=0} Q_i(z)λ^{k−i} \frac{d^i}{dz^i} , where each Q_i(z) is a polynomial of degree at most i and λ is the spectral parameter. We show that under mild nondegeneracy assumptions for all sufficiently large positive integers n there exist exactly k distinct values λ_{n,j} , 1 ≤ j ≤ k , of the spectral parameter λ such that the operator T_λ has a polynomial eigenfunction p_{n,j}(z) of degree n . These eigenfunctions split into k different families according to the asymptotic behavior of their eigenvalues. We conjecture and prove sequential versions of three fundamental properties: the limits Ψ_j (z) = \lim_{n→∞} \frac{p'_{n,j}(z)}{λ_{n,j}p_{n,j}(z)} exist, are analytic and satisfy the algebraic equation \sum^k_{i=0} Qi(z)Ψ^i_j(z) = 0 almost everywhere in ℂ ℙ^1 . As a consequence we obtain a class of algebraic functions possessing a branch near ∞ ∈ ℂ ℙ^1 which is representable as the Cauchy transform of a compactly supported probability measure.